Special hole elements for simulating the heat conduction in two-dimensional cellular materials

Abstract The hybrid Trefftz finite element method (HT-FEM) employs the T-complete solutions as domain approximation bases, providing a cheap boundary-type approach to simulate the heat conduction in two-dimensional cellular materials. The proposed method assumes two independent temperature fields inside the element and on its boundary. A conformal mapping function is introduced so that the T-complete solutions are established for the domain with a triangle- or maize-like hole. With the aid of the modified variational functional, the two assumed temperature fields are linked together at the element level and the conformity is enforced across inter-element boundaries. The resultant finite element formulation involves boundary integrals only, which makes a great flexibility in mesh generation with polygons. Numerical examples demonstrate that the proposed method can capture the local effects around the triangle- or maize-like holes precisely with extremely coarse meshes. Compared to ABAQUS, the proposed method exhibits higher convergence rate and computational efficiency.